Planar Algebra

Planar Algebra

In mathematics, planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a II₁ subfactor . They also provide an appropriate algebraic framework for many knot invariants (in particular the Jones polynomial), and have been used in describing the properties of Khovanov homology with respect to tangle composition .

Given a label set with an involution, and a fixed set of words in the elements of the label set, a planar algebra consists of a collection of modules, one for each element in, together with an action of the operad of tangles labelled by .